Graphene and layered materials in general exhibit rich physics and application potential owing to their exceptional electronic properties, which arise from the intricate π-orbital coupling and the symmetry breaking in twisted bilayer systems. Here, we report room-temperature experiments to study electrical transport across a bilayer graphene interface with a well-defined rotation angle between the layers that is controllable in situ. This twisted interface is artificially created in mesoscopic pillars made of highly oriented pyrolytic graphite by mechanical actuation. The overall measured angular dependence of the conductivity is consistent with a phonon-assisted transport mechanism that preserves the electron momentum of conduction electrons passing the interface. The most intriguing observations are sharp conductivity peaks at interlayer rotation angles of 21.8^o and 38.2^o. These angles correspond to a commensurate crystalline superstructure leading to a coherent two-dimensional (2D) electronic interface state. Such states, predicted by theory, form the basis for a new class of 2D weakly coupled bilayer systems with hitherto unexplored properties and applications.